Question about the representability of a relation

Question

I am trying to prove lemma 9.3,But I cant understand how if we have $T \vdash \phi(\bar{n_1},..\bar{n_r})$. Therefore $T\vdash \psi(\bar{n_1},...\bar{n_r},v) \iff v=\bar{0}$. I don't know how to prove this statement, it will be great if you can help me! Any help is appreiciated. Thanks!

Answer 1

I cant understand how if we have $T ⊢ \varphi(\overline n)$, therefore $T ⊢ \psi(\overline n,v) ⟺ v =\overline 0$ [simplified the number of arguments].

The formula $\psi$ has been defined- as: $(\varphi(\overline n) \land v=\overline 0) \lor (\lnot \varphi(\overline n) \land v=\overline 1)$.

Suppose that $P(n)$ holds, that means that $K_P(n)=0$.

By representability we have $T⊢ \varphi(\overline n)$ and thus the first disjunct of $\psi$ will hold iff $v=\overline 0$.

We have a predicate $P$ which is represented by formula $\varphi$, and we have the corresponding characteristic function $K_P$.

The gist of the Lemma is to define a new formula $\psi(n,v)$ such that if $P(n)$ holds, and thus $K_P(n)=0$, then $T \vdash \psi(n,0)$ and if $¬P(n)$ holds, then $T \vdash \psi(n,1)$.

If so, formula $\psi$ will represent the characteristic function $K_P$.